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Q: For more than a decade, the Berkshire-based company has used the address to direct shoppers to a Star Wars section of its Jokers' Masquerade store.
But in July, Nominet, which oversees .uk domains, backed Disney's ownership claim.
The last time anyone successfully appealed against a Nominet ruling was in 2013.
The costume store's parent company, Abscissa, was also told to give up a further six domain names used for the same purpose:
Chief executive Mark Lewis said Abscissa had used two of the addresses for more than 12 years without being challenged.
"I can't believe that over the last two decades that someone from either Lucasfilm or Disney did not do a WhoIs [search] and find that that starwars.co.uk and star-wars.co.uk were not registered to them," he told the BBC.
"There has to be a point in time, surely, where a registrant has to be able to hold some title."
He added that Lucasfilm had owned star-wars.co.uk for a time prior to 2003, but had chosen not to renew it.
"We cannot find any case where a complainant lets a domain lapse, then files a complaint," he said. "I believe this case sets a precedent."
Nominet requires a complainant to prove that a domain name registration is "abusive" for it to agree to transfer ownership.
Its initial ruling supported Disney's claim on the basis that consumers visiting the sites would have "falsely inferred a commercial connection" between the fancy dress store and the film franchise.
But Mr Lewis disputes this conclusion.
"We haven't abused them," he said.
"We haven't rented them, we haven't offered them for sale - the internet domains point to legitimate Star Wars-branded costumes that we've been selling for the past 13 years."
He added that he did not believe the two sides would be engaged in the legal battle at all had Nominet not introduced shorter "name.uk" domains last year.
Disney bought Lucasfilm - the production company behind Star Wars - in 2012 for $4.1bn (Â£2.6bn). It plans to release a new film - The Force Awakens - in December.
A spokesman for the entertainment giant could not be reached for comment.
Nominet's initial rulings are determined by single expert, but appeals go to a three-person panel.
Since 2001, just 48 cases - representing 1.8% of the body's rulings - have led to an appeal. Of those appeals, 20 resulted in the original decision being overturned.
A spokesman for Nominet said it was unable to comment on the Star Wars case while it remained active.
A:
A fancy-dress retailer is appealing against a ruling it must surrender its starwars.co.uk web address to Disney.